Sequential Methods for Coupled Geomechanics and Multiphase Flow

2.3. COUPLING AND CONSTITUTIVE RELATIONS 19
Differentiating Equation 2.13,
δσ = ∂2Ψ ∂2ε : δε + ∂2Ψ δmJ +
∂ε∂mJ
∂2Ψ δT,
∂ε∂T
(2.14)
δgJ =
∂2Ψ ∂ε∂mJ
: δε + ∂2Ψ δmK +
∂mJ∂mK
∂2Ψ δT,
∂mJ∂T
(2.15)
where δgJ= 1
δp + sJδT,
ρJ
δS = ∂2Ψ ∂ε∂T : δε + ∂2Ψ ∂mJ∂T δmJ + ∂2Ψ ∂2 δT, (2.16)
T
where δ represents the incremental form of the material derivative with respect to the solid
skeleton, d (·)/dt. Then, the constitutive relations for elasticity are obtained as follows:
where
∂ 2 Ψ
(∂m/ρ) J (∂m/ρ) K
δm
δσ = Cud : δε − MJKbK − AqδT , (2.17)
ρ J
δm
δpJ = MJK −bK : δε + + (ρlq)
ρ
JδT , (2.18)
K
δm
δS = Aq : δε − lq,J + βqδT , (2.19)
ρ J
∂2Ψ ∂2ε = Cud
∂
,
2Ψ = −MJKbK ,
∂ε(∂m/ρ) J
∂2Ψ ∂ε∂T = −Aq ,
∂
= MJK ,
2
Ψ
− sJ = lq,J ,
(∂m/ρ) J ∂T ∂2Ψ ∂2T = βq, (2.20)
where Cud (undrained moduli), MJK (Biot moduli), bK (Biot coefficient), Aq, lq,J, and βq
are determined from experiments. bK is the second-order tensor, and becomes bK1 in the
isotropic case, where 1 is the second-order identity tensor. When isotropic and isothermal
conditions are assumed, the constitutive relations of δσ, δpJ, and δS are reduced to the